Existence and Uniqueness of Solutions: Example 1
Example: Suppose the differential equation satisfies the Existence and Uniqueness Theorem
for all values of y and t. Suppose and
are two solutions to this differential equation.
- What can you say about the behavior of the solution of the
solution y(t) satisfying the initial condition y(0)=1?
Hint: Draw the two solutions and .
- Address the behavior of y(t) as t approaches ,
and as t approaches .
- First let us draw the graphs of and .
Since we have , we deduce from the
Existence and Uniqueness Theorem that for all t, we have
In particular, y(t) has the line y=t as an oblique asymptote
which answers the second question.
We cannot predict that y(t) is an increasing function.
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